Total outer-connected domination in trees

Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by ${\displaystyle {\gamma }_{tc}\left(G\right)}$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then ${\displaystyle {\gamma }_{tc}\left(T\right)\ge ⎡2\frac{n}{3}⎤}$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.